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Would it suffice to divide by the size of the automorphism group of the subgraph that is being searched for, perhaps? In other words, are you trying to count non labelled occurrences of cycles? For the particular case stated in your question, dividing by 8 might do the trick.

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Seems like you are searching for non labelled occurrences of cycles. In such case dividing by the size of the automorphism group of the graph that is being searched for might do the trick. Say you are trying to count the number of unlabelled appearances of graph H in graph G. You can define